In this article, we consider computing expectations w.r.t. probability measures which are subject to discretization error. Examples include partially observed diffusion processes or inverse problems, where one may have to discretize time and/or space in order to practically work with the probability of interest. Given access only to these discretizations, we consider the construction of unbiased Monte Carlo estimators of expectations w.r.t. such target probability distributions. It is shown how to obtain such estimators using a novel adaptation of randomization schemes and Markov simulation methods. Under appropriate assumptions, these estimators possess finite variance and finite expected cost. There are two important consequences of this approach: (i) unbiased inference is achieved at the canonical complexity rate, and (ii) the resulting estimators can be generated independently, thereby allowing strong scaling to arbitrarily many parallel processors. Several algorithms are presented and applied to some examples of Bayesian inference problems with both simulated and real observed data.
Bibliographical noteKAUST Repository Item: Exported on 2023-06-07
Acknowledgements: We thank two referees and an associate editor for extremely useful comments that have greatly improved the paper. The work of the first author was funded by CY Initiative of Excellence (grant ``Investissements d'Avenir""ANR-16-IDEX-0008). The work of the second author was supported by KAUST baseline funding. The work of thethird and fourth authors was supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.